Question

derivada de
4csc(x)^2 +3cot(x)^2 - 4cot(x) - 7​

Answer 1

Respuesta:     $4\csc ^2\left(x\right)-14\csc ^2\left(x\right)\cot \left(x\right)$

Explicación paso a paso:

$\frac{d}{dx}\left(4\csc ^2\left(x\right)+3\cot ^2\left(x\right)-4\cot \left(x\right)-7\right)$

$\mathrm{Aplicar\:la\:regla\:de\:la\:suma/diferencia}:\quad \left(f\pm g\right)'=f\:'\pm g'$

$=\frac{d}{dx}\left(4\csc ^2\left(x\right)\right)+\frac{d}{dx}\left(3\cot ^2\left(x\right)\right)-\frac{d}{dx}\left(4\cot \left(x\right)\right)-\frac{d}{dx}\left(7\right)$

Desarrollar:  $\frac{d}{dx}\left(4\csc ^2\left(x\right)\right)$  :  $\mathrm{Sacar\:la\:constante}:\quad \left(a\cdot f\right)'=a\cdot f\:'$

$=4\frac{d}{dx}\left(\csc ^2\left(x\right)\right)$

$\mathrm{Aplicar\:la\:regla\:de\:la\:cadena}:\quad \frac{df\left(u\right)}{dx}=\frac{df}{du}\cdot \frac{du}{dx}$

$f=u^2,\:\:u=\csc \left(x\right)$

$=4\frac{d}{du}\left(u^2\right)\frac{d}{dx}\left(\csc\left(x\right)\right)$

$=4\cdot \:2u\left(-\cot \left(x\right)\csc \left(x\right)\right)$

$\mathrm{Sustituir\:en\:la\:ecuacion}\:u=\csc \left(x\right)$

$=4\cdot \:2\csc \left(x\right)\left(-\cot \left(x\right)\csc \left(x\right)\right)$

$=-8\csc ^2\left(x\right)\cot \left(x\right)$

Queda:

$=-8\csc ^2\left(x\right)\cot \left(x\right)+\frac{d}{dx}\left(3\cot ^2\left(x\right)\right)-\frac{d}{dx}\left(4\cot \left(x\right)\right)-\frac{d}{dx}\left(7\right)$

Desarrollar:  $\frac{d}{dx}\left(3\cot ^2\left(x\right)\right)$ : $\mathrm{Sacar\:la\:constante}:\quad \left(a\cdot f\right)'=a\cdot f\:'$

$=3\frac{d}{dx}\left(\cot ^2\left(x\right)\right)$

$\mathrm{Aplicar\:la\:regla\:de\:la\:cadena}:\quad \frac{df\left(u\right)}{dx}=\frac{df}{du}\cdot \frac{du}{dx}$

$f=u^2,\:\:u=\cot \left(x\right)$

$=3\frac{d}{du}\left(u^2\right)\frac{d}{dx}\left(\cot \left(x\right)\right)$

$=3\cdot \:2u\left(-\csc ^2\left(x\right)\right)$

$\mathrm{Sustituir\:en\:la\:ecuacion}\:u=\cot \left(x\right)$

$=3\cdot \:2\cot \left(x\right)\left(-\csc ^2\left(x\right)\right)$

$\mathrm{Simplificar}$

$=-6\csc ^2\left(x\right)\cot \left(x\right)$

Queda:

$=-8\csc ^2\left(x\right)\cot \left(x\right)-6\csc ^2\left(x\right)\cot \left(x\right)-\frac{d}{dx}\left(4\cot \left(x\right)\right)-\frac{d}{dx}\left(7\right)$

Desarrollar:  $\frac{d}{dx}\left(4\cot \left(x\right)\right)$   : $\mathrm{Sacar\:la\:constante}:\quad \left(a\cdot f\right)'=a\cdot f\:'$

$=4\frac{d}{dx}\left(\cot \left(x\right)\right)$

$\mathrm{Aplicar\:la\:regla\:de\:derivacion}:\quad \frac{d}{dx}\left(\cot \left(x\right)\right)=-\csc ^2\left(x\right)$

$=4\left(-\csc ^2\left(x\right)\right)$

$\mathrm{Simplificar}$

$=-4\csc ^2\left(x\right)$

Queda:

$=-8\csc ^2\left(x\right)\cot\left(x\right)-6\csc ^2\left(x\right)\cot \left(x\right)-\left(-4\csc ^2\left(x\right)\right)-\frac{d}{dx}\left(7\right)$

Desarrollar: $\frac{d}{dx}\left(7\right)$  : $\mathrm{Derivada\:de\:una\:constante}:\quad \frac{d}{dx}\left(a\right)=0$

$=0$

Queda finalmente :

$=-8\csc ^2\left(x\right)\cot \left(x\right)-6\csc ^2\left(x\right)\cot \left(x\right)-\left(-4\csc ^2\left(x\right)\right)-0$

$\mathrm{Aplicar\:la\:regla}\:-\left(-a\right)=a$

$=-8\csc ^2\left(x\right)\cot \left(x\right)-6\csc ^2\left(x\right)\cot \left(x\right)+4\csc ^2\left(x\right)-0$$\mathrm{Sumar\:elementos\:similares:}\:-8\csc ^2\left(x\right)\cot \left(x\right)-6\csc ^2\left(x\right)\cot \left(x\right)=-14\csc ^2\left(x\right)\cot \left(x\right)$$=-14\csc ^2\left(x\right)\cot \left(x\right)+4\csc ^2\left(x\right)-0$

$14\csc ^2\left(x\right)\cot \left(x\right)+4\csc ^2\left(x\right)-0=-14\csc ^2\left(x\right)\cot \left(x\right)+4\csc ^2\left(x\right)$

$=4\csc ^2\left(x\right)-14\csc ^2\left(x\right)\cot \left(x\right)$   ⇒ Solución